In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces which are complete with respect to the metric induced by the norm). Fréchet spaces, in contrast, are locally convex spaces which are complete with respect to a translation invariant metric.
Even though the topological structure of Fréchet spaces is more complicated than that of Banach spaces due to the lack of a norm, many important results in functional analysis, like the open mapping theorem and the BanachSteinhaus theorem, still hold.
Spaces of infinitely differentiable functions are typical examples of Fréchet spaces.
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Fréchet spaces can be defined in two equivalent ways: the first employs a translationinvariant metric, the second a countable family of seminorms.
A topological vector space X is a Fréchet space if and only if it satisfies the following three properties:
Note that there is no natural notion of distance between two points of a Fréchet space: many different translationinvariant metrics may induce the same topology.
The alternative and somewhat more practical definition is the following: a topological vector space X is a Fréchet space if and only if it satisfies the following three properties:
A sequence (x_{n}) in X converges to x in the Fréchet space defined by a family of seminorms if and only if it converges to x with respect to each of the given seminorms.
To construct a Fréchet space, one typically starts with a vector space X and defines a countable family of seminorms ._{k} on X with the following two properties:
Then the topology induced by these seminorms (as explained above) turns X into a Fréchet space; the first property ensures that it is Hausdorff, and the second property ensures that it is complete. A translationinvariant complete metric inducing the same topology on X can then be defined by
Note that the function u → u / (1+u) maps [0, ∞) monotonically to [0, 1), and so the above definition ensures that d(x, y) is "small" if and only if there exists K "large" such that x − y_{k} is "small" for k = 0, …, K.
Trivially, every Banach space is a Fréchet space as the norm induces a translation invariant metric and the space is complete with respect to this metric.
The vector space C^{∞}([0,1]) of all infinitely often differentiable functions ƒ : [0, 1] → R becomes a Fréchet space with the seminorms
for every integer k ≥ 0. Here, ƒ^{ (k)} denotes the kth derivative of ƒ, and ƒ^{ (0)} = ƒ.
In this Fréchet space, a sequence (ƒ_{n}) of functions converges towards the element ƒ of C^{∞}([0, 1]) if and only if for every integer k ≥ 0, the sequence (ƒ_{n}^{(k)}) converges uniformly towards ƒ^{ (k)}.
Similarly, the vector space C^{∞}(R) of all infinitely often differentiable functions ƒ : R → R becomes a Fréchet space with the seminorms
for all integers k, n ≥ 0.
The vector space C^{m}(R) of all mtimes continuously differentiable functions ƒ : R → R becomes a Fréchet space with the seminorms
for all integers n ≥ 0 and k=0,...,m.
More generally, if M is a compact C^{∞}manifold and B is a Banach space, then the set C^{∞}(M,B) of all infinitelyoften differentiable functions ƒ : M → B can be turned into a Fréchet space by using as seminorms the suprema of the norms of all partial derivatives. If M is a (not necessarily compact) C^{∞}manifold which admits a countable sequence K_{n} of compact subsets, so that every compact subset of M is contained in at least one K_{n}, then the spaces C^{m}(M,B) and C^{∞}(M,B) are also Fréchet space in a natural manner.
The space R^{ω} of all real valued sequences becomes a Fréchet space if we define the kth seminorm of a sequence to be the absolute value of the kth element of the sequence. Convergence in this Fréchet space is equivalent to elementwise convergence.
Not all vector spaces with complete translationinvariant metrics are Fréchet spaces. An example is the space L^{p}([0, 1]) with p < 1. This space fails to be locally convex. It is a Fspace.
Several important tools of functional analysis which are based on the Baire category theorem remain true in Fréchet spaces; examples are the closed graph theorem and the open mapping theorem.
If X and Y are Fréchet spaces, then the space L(X,Y) consisting of all continuous linear maps from X to Y is not a Fréchet space in any natural manner. This is a major difference between the theory of Banach spaces and that of Fréchet spaces and necessitates a different definition for continuous differentiability of functions defined on Fréchet spaces, the Gâteaux derivative:
Suppose X and Y are Fréchet spaces, U is an open subset of X, P : U → Y is a function, x ∈ U and h ∈ X. We say that P is differentiable at x in the direction h if the limit
exists. We call P continuously differentiable in U if
is continuous. Since the product of Fréchet spaces is again a Fréchet space, we can then try to differentiate D(P) and define the higher derivatives of P in this fashion.
The derivative operator P : C^{∞}([0,1]) → C^{∞}([0,1]) defined by P(f) = f′ is itself infinitely differentiable. The first derivative is given by
for any two elements f and h in C^{∞}([0,1]). This is a major advantage of the Fréchet space C^{∞}([0,1]) over the Banach space C^{k}([0,1]) for finite k.
If P : U → Y is a continuously differentiable function, then the differential equation
need not have any solutions, and even if does, the solutions need not be unique. This is in stark contrast to the situation in Banach spaces.
The inverse function theorem is not true in Fréchet spaces; a partial substitute is the Nash–Moser theorem.
One may define Fréchet manifolds as spaces that "locally look like" Fréchet spaces (just like ordinary manifolds are defined as spaces that locally look like Euclidean space R^{n}), and one can then extend the concept of Lie group to these manifolds. This is useful because for a given (ordinary) compact C^{∞} manifold M, the set of all C^{∞} diffeomorphisms f : M → M forms a generalized Lie group in this sense, and this Lie group captures the symmetries of M. Some of the relations between Lie algebras and Lie groups remain valid in this setting.
If we drop the requirement for the space to be locally convex, we obtain Fspaces: vector spaces with complete translationinvariant metrics.
LFspaces are countable inductive limits of Fréchet spaces.
